Lectures for Part II: Time Series Models in Finance
Lectures for Part II: Time Series Models in Finance
Lectures for Part II: Time Series Models in Finance
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Hill’s estimator of tail <strong>in</strong>dex<br />
Def: The Hill estimate of α <strong>for</strong> heavy-tailed data with distribution given<br />
by<br />
1-F(x) = x -α L(x),<br />
is<br />
αˆ<br />
−1<br />
=<br />
1<br />
m<br />
m<br />
∑( log X<br />
( j)<br />
− log X<br />
( j+<br />
1)<br />
)<br />
j=<br />
1<br />
j<br />
=<br />
1<br />
m<br />
m<br />
∑( log X<br />
( j)<br />
− log X<br />
( m+<br />
1)<br />
)<br />
j=<br />
1<br />
The asymptotic variance of this estimate <strong>for</strong> α is<br />
ˆ 2<br />
2<br />
α / m and estimated by α / m.<br />
(See also GPD=generalized Pareto distribution.)<br />
MaPhySto Workshop 9/04<br />
24
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